Some Mean Convergence Theorems for the Maximum of Normed Double Sums of Banach Space Valued Random Elements

In this correspondence,we establish mean convergence theorems for the maximum of normed double sums of Banach space valued random elements.Most of the results pertain to random elements which are M-dependent.We expand and improve a number of particular cases in the literature on mean convergence of random elements in Banach spaces.One of the main contributions of the paper is to simplify and improve a recent result of Li,Presnell,and Rosalsky[Journal of Mathematical Inequalities,16,117–126(2022)].A new maximal inequality for double sums of M-dependent random elements is proved which may be of independent interest.The sharpness of the results is illustrated by four examples.

The space time CE/SE method for solving one-dimensional batch crystallization model with fines dissolution

This article is concerned with the numerical investigation of one-dimensional population balance models for batch crystallization process with fines dissolution.In batch crystallization,dissolution of smaller unwanted nuclei below some critical size is of vital importance as it improves the quality of product.The crystal growth rates for both size-independent and size-dependent cases are considered.A delay in recycle pipe is also included in the model.The space–time conservation element and solution element method,originally derived for non-reacting flows,is used to solve the model.This scheme has already been applied to a range of PDEs,mainly in the area of fluid mechanics.The numerical results are compared with those obtained from the Koren scheme,showing that the proposed scheme is more efficient.

ON ALMOST SURE CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENT SEQUENCES

We mainly study the almost sure limiting behavior of weighted sums of the form ∑ni=1 aiXi/bn , where {Xn, n ≥ 1} is an arbitrary Banach space valued random element sequence or Banach space valued martingale difference sequence and {an, n ≥ 1} and {bn,n ≥ 1} are two sequences of positive constants. Some new strong laws of large numbers for such weighted sums are proved under mild conditions.

A NEW FINITE ELEMENT SPACE FOR EXPANDED MIXED FINITE ELEMENT METHOD

In this article,we propose a new finite element spaceΛh for the expanded mixed finite element method(EMFEM)for second-order elliptic problems to guarantee its computing capability and reduce the computation cost.The new finite element spaceΛh is designed in such a way that the strong requirement V h⊂Λh in[9]is weakened to{v h∈V h;d i v v h=0}⊂Λh so that it needs fewer degrees of freedom than its classical counterpart.Furthermore,the newΛh coupled with the Raviart-Thomas space satisfies the inf-sup condition,which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix,and thus the existence,uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in R d,d=2,3 and for triangular partitions in R 2.Also,the solvability of the EMFEM for triangular partition in R 3 can be directly proved without the inf-sup condition.Numerical experiments are conducted to confirm these theoretical findings.

A Development of Rom-Exercise Machine Using Linkages(Design of Leg Mechanism by Walking Simulation)

In this study, the design of an automatic ROM-Exercise machine that is constructed witha planar multililnk mechamsm consisting of only revolute pairs is investigated. Namely, equations thatdetermine the minimum moving spaces and relative positions of link required to construct the legmechanism are formulated with consideration of transform functions. For the leg mechanism that isconstructed with a planar eleven-link mechanism, arrangements of each link and optimum linkprofiles avoided mutual interferences among moving links are determined wb consideration of therelative locations of each link in the same plane during a cycle of motion of the mechanism. Based onthe above analytical results, an automatic ROM-Exercise machine that performs within a minimum moving spaces is proposed as a prachcal example. ms study is carried out as part of the students'computcr education to the graduation thesis, in order to improve their creativity and machine designtechnology skills in coniunction with educational advantages. Significam educational results areobtained by using the design techniques mentioned above.

Some Strong Laws of Large Numbers for Blockwise Martingale Difference Sequences in Martingale Type p Banach Spaces

For a blockwise martingale difference sequence of random elements {Vn, n ≥ 1} taking values in a real separable martingale type p （1 ≤ p ≤ 2） Banach space, conditions are provided for strong laws of large numbers of the form limn→∞ Vi/gn = 0 almost surely to hold where the constants gn ↑∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p （1〈 p ≤2） Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.

Stiffness Matrix Derivation of Space Beam Element at Elevated Temperature

Element stiffness equation is very important in structural analysis, and directly influences the accuracy of the results. At present, derivation method of element stiffness equation is relatively mature under ambient temperature, and the elastic phrase of material stress-strain curve is generally adopted as physical equation in derivation. However, the material stress-strain relationship is very complicated at elevated temperature, and its form is not unique, which brings great difficulty to the derivation of element stiffness equation. Referring to the derivation method of element stiffness equation at ambient temperature, by using the continuous function of stress-strain-temperature at elevated temperature, and based on the principle of virtual work, the stiffness equation of space beam element and the formulas of stiffness matrix are derived in this paper, which provide basis for finite element analysis on structures at elevated temperature.

A DISCONTINUOUS GALERKIN METHOD FOR THE FOURTH-ORDER CURL PROBLEM

In this paper, we present a discontinuous Galerkin （DG） method based on the N@d@lec finite element space for solving a fourth-order curl equation arising from a magnetohy- drodynamics model on a 3-dimensional bounded Lipschitz polyhedron. We show that the method has an optimal error estimate for a model problem involving a fourth-order curl operator. Furthermore, some numerical results in 2 dimensions are presented to verify the theoretical results.

Elevation spatial characteristic investigation based on 3D massive MIMO channel measurements at 6 GHz

As the key technology of the 5th generation （5G）, 3-dimensional （3D） massive multi-input and multi-output （MIMO） is expected to be widely used in small cell network （SCN）. In this paper, in order to investigated the tradeoff between limited size in SCN and the capacity gain from increasing antenna elements, the spatial performances of 3D massive MIMO based on a 512×16 MIMO channel measurements at 6 GHz in urban microcell （UMi） scenario are studied. Enormous channel impulse responses （CIR） are collected and reconstructed, which enables us to present comparative results of the capacity and the eigenvalue spread （ES）. Furthermore, the impacts of antenna element number and spacing on system performance are investigated, i. e. , 32, 64, 128 elements are selected from the 512 transmitter （Tx） array with elevation interval spacing being 0.5, 1 and 2 wavelengths for each. Interestingly, the capacity gap can be obviously observed on the comparison between the 1 and 2 wavelength antenna spacing cases, which implies that correlation cannot be ignored when the antenna spacing is larger than 1 wavelength when massive antennas are equipped. The contrast results show that the capacities are enlarged with the increasing of antenna elements number, and larger antenna spacing will lead to higher channel capacity as expected. However, the capacity gains brought by the increasing of antenna spacing will descend to certain degrees as the antenna number increases. Collectively, these results will provide further insights into 3D massive MIMO utilization.

INTERIOR SUPERCONVERGENCE ERRORESTIMATES FOR MIXED FINITEELEMENT METHODS FOR SECOND ORDER ELLIPTIC PROBLEM

The aim of this paper is to provide a local superconvergence analysis for ined finite element methods of Poission equation. We shall prove that if p is smmoth enough in a local regionΩ0Ω1Ω and rectangular mesh is imposed onΩ1, then local superconvergence for are expected. Thus, by post-processing operators P and we have obtained the follwing local superconvergence error estimate: